This series of papers aims at understanding the formation and evolution of non-barred disc galaxies. We use the new spectro-photometric decomposition code, C2D, to separate the spectral information of bulges and discs of a statistically representative sample of galaxies from the CALIFA survey. Then, we study their stellar population properties analising the structure-independent datacubes with the Pipe3D algorithm. We find a correlation between the bulge-to-total ($B/T$) luminosity (and mass) ratio and galaxy stellar mass. The $B/T$ mass ratio has only a mild evolution with redshift, but the bulge-to-disc ($B/D$) mass ratio shows a clear increase of the disc component since redshift $z < 1$ for massive galaxies. The mass-size relation for both bulges and discs describes an upturn at high galaxy stellar masses (log{(M_{star}/M_{sun})} > 10.5). The relation holds for bulges but not for discs when using their individual stellar masses. We find a negligible evolution of the mass-size relation for both the most massive (log{(M_{star rm ,b,d}/M_{sun})} > 10) bulges and discs. For lower masses, discs show a larger variation than bulges. We also find a correlation between the Sersic index of bulges and both galaxy and bulge stellar mass, which does not hold for the disc mass. Our results support an inside-out formation of nearby non-barred galaxies, and they suggest that i) bulges formed early-on and ii) they have not evolved much through cosmic time. However, we find that the early properties of bulges drive the future evolution of the galaxy as a whole, and particularly the properties of the discs that eventually form around them.
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